Problem: Compute the smallest positive angle $x,$ in degrees, such that
\[\tan 4x = \frac{\cos x - \sin x}{\cos x + \sin x}.\]
Solution: From the given equation,
\[\frac{\sin 4x}{\cos 4x} = \frac{\cos x - \sin x}{\cos x + \sin x}.\]Then
\[\cos x \sin 4x + \sin x \sin 4x = \cos x \cos 4x - \sin x \cos 4x,\]or
\[\cos x \sin 4x + \sin x \cos 4x = \cos x \cos 4x - \sin x \sin 4x.\]Applying sum-to-product to both sides, we get
\[\sin 5x = \cos 5x,\]so $\tan 5x = 1.$  The smallest such positive angle $x$ is $\boxed{9^\circ}.$